Ergodic Theory, Groups, and Geometry: Nsf-cbms Regional Research Conferences in the Mathematical Sciences June 22-26, 1998 University of Minnesota

by Robert J. Zimmer, Dave Witte Morris

The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure theoretic techniques. The lecture notes collected here provide an introduction to some of the important methods, major developments, and open problems in the subject. They are slightly expanded from lectures given by the author at a 1998 CBMS Conference held at the University of Minnesota. The main text presents a perspective on the field as it was 10 years ago; comments after the notes of each lecture provide suggestions for further reading, including references to recent developments. Some topics examined are superrigidity and first applications, locally homogeneous spaces, and orbit equivalence. An appendix of background material is included. Annotation ©2009 Book News, Inc., Portland, OR (booknews.com)

85 pages, Paperback

First published December 12, 2008

Reviews

[Kevin K. Gillette · Rating 3 out of 5]

Though this is an outstanding monograph, I confess that I was a little misled by the title. It leads with ergodic theory, but this theory (and in particular, Moore's Ergodic Theorem) doesn't eevn show up until almost midway through the text. Rather, this book is far more about the topological and algebraic relationships between several classes of manifolds, group actions, and homomorphisms and isomorphisms that conflate and characterize the manifolds.

This is definitely heavy-duty math, so those light on their group theory, manifold theory, differential geometry and modern algebra are advised to tread cautiously. But interesting nuggets await anyone patient enough to wade through the morass of definitions and theorems/corollaries/lemmas/propositions (and there are plenty of those!).

[Woflmao · Rating 3 out of 5]

This book is essentially a long survey article on the actions of semisimple Lie groups and their discrete subgroups on measure spaces. As such, it has no pedagogical angle, but rather communicates the current results in a very economical manner. As a survey it works well in giving the reader a feel for what is going on in this area, but it will leave no non-expert quite puzzled most of the time. Still, due to comprehensive and annotated literature lists, one is well-prepared to delve deeper into the matter on its own.